Inverse Sturm-Liouville Problem Research Articles

Solving inverse nodal problem with spectral parameter in boundary conditions

ABSTRACTIn this study, the inverse Sturm-Liouville problem with spectral parameter in boundary conditions on a finite interval is considered. In fact, we use the nodes as input data and compute the approximation of solution of the inverse nodal Sturm-Liouville problem by the first kind Chebyshev polynomials and apply two methods Chebyshev wavelets and Chebyshev interpolation for solving inverse nodal Sturm-Liouville problem. Finally, the results are explained by presenting the numerical example.

On a method for solving the inverse Sturm–Liouville problem

Abstract A method for solving the inverse Sturm–Liouville problem on a finite interval is proposed. It is based on a Fourier–Legendre series representation of the integral transmutation kernel. Substitution of the representation into the Gel’fand–Levitan equation leads to a linear algebraic system of equations and consequently to a simple algorithm for recovering the potential. Numerical illustrations are presented.

On the solution of an inverse Sturm–Liouville problem with a delay andeigenparameter-dependent boundary conditions

In this paper, a boundary value problem consisting of a delay differential equation of the Sturm-Liouville type with eigenparameter-dependent boundary conditions is investigated. The asymptotic behavior of eigenvalues is studied and the parameter of delay is determined by eigenvalues. Then we obtain the connection between the potential function and the canonical form of the characteristic function.

On the Uniqueness of the Solution of the Inverse Sturm–Liouville Problem with Nonseparated Boundary Conditions on a Geometric Graph

For the first time, the inverse Sturm–Liouville problem with nonseparated boundary conditions is studied on a star-shaped geometric graph with three edges. It is shown that the Sturm–Liouville problem with general boundary conditions cannot be uniquely reconstructed from four spectra. Nonseparated boundary conditions are found for which a uniqueness theorem for the solution of the inverse Sturm–Liouville problem is proved. The spectrum of the boundary value problem itself and the spectra of three auxiliary problems are used as reconstruction data. It is also shown that the Sturm–Liouville problem with these nonseparated boundary conditions can be uniquely recovered if three spectra of auxiliary problems are used as reconstruction data and only five of its eigenvalues are used instead of the entire spectrum of the problem.

On the Uniqueness of the Solution of the Inverse Sturm-Liouville Problem with Nonseparated Boundary Conditions on a Geometric Graph

On the Uniqueness of the Solution of the Inverse Sturm-Liouville Problem with Nonseparated Boundary Conditions on a Geometric Graph

Inverse Sturm-Liouville problem for a star graph by three spectra

Inverse Sturm-Liouville problem for a star graph by three spectra

Solving inverse Sturm-Liouville problems: theory and practice

Theoretical results on the solution of inverse Sturm-Liouville problems generally consider only idealized problems requiring much more data than is available in real applications. Typical theorems describe problems where infinitely many eigenvalues are known exactly, but in most applications we know only approximations of a finite, and usually small, number of eigenvalues. This paper considers how idealized theoretical results may assist practical numerical computation. It also reviews recent progress on a class of numerical methods for inverse Sturm-Liouville problems, it discusses some open questions, and it announces a new convergence result. References L. Aceto, P. Ghelardoni and C. Magherini. Boundary value methods for the reconstruction of Sturm–Liouville potentials. Appl. Math. Comp. , 219 :2960–2974, 2012. doi:10.1016/j.amc.2012.09.021 A. M. Akhtyamov, V. A. Sadovnichy and Ya. T. Sultanaev. Generalizations of Borg's uniqueness theorem to the case of non- separated boundary conditions. Eurasian Math. J. , 3(4) :10–22, 2012. http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=emj&paperid=101&option_lang=eng H. Altundag, C. Bockmann and H. Taseli. Inverse Sturm–Liouville problems with pseudospectral methods. Int. J. Comput. Math. , 92 :1373–1384, 2015. doi:10.1080/00207160.2014.939646 A. L. Andrew. Asymptotic correction of Numerov's eigenvalue estimates with natural boundary conditions. J. Comp. Appl. Math. , 125 :359–366, 2000. doi:10.1016/S0377-0427(00)00479-9 A. L. Andrew. Numerov's method for inverse Sturm–Liouville problems. Inverse Problems , 21 :223–238, 2005. doi:10.1088/0266-5611/21/1/014 A. L. Andrew. Asymptotic correction and inverse eigenvalue problems: an overview. ANZIAM J. , 46(E) :C1–C14, 2005. doi:10.21914/anziamj.v46i0.943 A. L. Andrew. Computing Sturm–Liouville potentials from two spectra. Inverse Problems 22 :2069–2081, 2006. doi:10.1088/0266-5611/22/6/010 A. L. Andrew. Finite difference methods for half inverse Sturm–Liouville problems. Appl. Math. Comp. , 218 :445–457, 2011. doi:10.1016/j.amc.2011.05.085 A. L. Andrew. Inverse Sturm–Liouville problems: some recent developments. ANZIAM J. , 52(E) :C287–C302, 2011. doi:10.21914/anziamj.v52i0.3888 A. L. Andrew. Convergence of Numerov's method for an inverse Sturm–Liouville problem. In preparation. A. L. Andrew and J. W. Paine. Correction of Numerov's eigenvalue estimates. Numer. Math. , 47 :289–300, 1985. MR 86j:65101. doi:10.1007/BF01389712 L. Efremova and G. Freiling. Numerical solution of inverse problems for Sturm–Liouville operators with discontinuous potentials. Cent. Eur. J. Math. 11 :2044–2051, 2013. doi:10.2478/s11533-013-0301-1 Q. Gao. Decent [descent] flow methods for inverse Sturm–Liouville problem. Appl. Math. Modelling 36 :4452–4465, 2012. doi:10.1016/j.apm.2011.11.070 Q. Gao, X. Cheng and Z. Huang. Modified Numerov's method for inverse Sturm–Liouville problems. J. Comp. Appl. Math. 253 :181–199, 2013. doi:10.1016/j.cam.2013.04.025 Q. Gao, X. Cheng and Z. Huang. On a boundary value method for computing Sturm–Liouville potentials from two spectra. Internat. J. Comp. Math. 91 :490–513, 2014. doi:10.1080/00207160.2013.790535 Q. Gao, Z. Huang and A. Cheng. A finite difference method for an inverse Sturm–Liouville problem in impedance form. Numer. Algorithms 70 :669–690, 2015. doi:10.1007/s11075-015-9968-7 Q. Gao, Q. Zhao, X. Zheng and Y. Liung. Convergence of Numerov's method for inverse Sturm–Liouville problems. Appl. Math. Comp. , 293 :1–17, 2017. doi:10.1016/j.amc.2016.08.007 O. H. Hald. The inverse Sturm–Liouville problem and the Rayleigh–Ritz method. Math. Comp. , 32 :687–705, 1978. doi:10.1090/S0025-5718-1978-0501963-2 C.-S. Liu. Solving an inverse Sturm–Liouville problem by a Lie-group method. Bound. Value Probl. 2008 :749865, 2008. doi:10.1155/2008/749865 C.-S. Liu and S. N. Atluri. A novel fictitious time integration method for solving the discretized inverse Sturm–Liouville problems, for specified eigenvalues. CMES Comput. Model. Eng. Sci. 36 :261–285, 2008. doi:10.3970/cmes.2008.036.261 B. D. Lowe, M. Pilant and W. Rundell. The recovery of potentials from finite spectral data. SIAM J. Math. Anal. 23 :482–504, 1992. doi:10.1137/0523023 J. W. Paine, F. R. de Hoog and R. S. Anderssen. On the correction of finite difference eigenvalue approximations. Computing 26 :123–139, 1981. doi:10.1007/BF02241779 J. D. Pryce. Numerical solution of Sturm–Liouville problems. Oxford University Press, 1993. M. Rafler and C. Bockmann. Reconstructive method for inverse Sturm–Liouville problems with discontinuous potentials. Inverse Problems 23 :933–946, 2007. doi:10.1088/0266-5611/23/3/006 A. Rattana and C. Bockmann. Matrix methods for computing eigenvalues of Sturm–Liouville problems of order four. J. Comp. Appl. Math. 249 :144–156, 2013. doi:10.1016/j.cam.2013.02.024 W. Rundell and P. E. Sacks. Reconstruction techniques for classical Sturm–Liouville problems. Math. Comp. 58 :161–183, 1992. doi:10.1090/S0025-5718-1992-1106979-0 A. M. Savchuk. Reconstruction of the potential of the Sturm–Liouville operator from a finite set of eigenvalues and normalizing constants. Math. Notes 99 :715–228, 2016. doi:10.1134/S0001434616050102 A. M. Savchuk and A. A. Shkalikov. On the eigenvalues of the Sturm–Liouville operator with potentials from Sobolev spaces. Math. Notes 80 :814–832, 2006. doi:10.1007/s11006-006-0204-6

Solving inverse Sturm–Liouville problem with separated boundary conditions by using two different input data

ABSTRACTIn this study, we consider Sturm–Liouville (SL) equation under the separated boundary conditions on a finite interval. We get the approximate solutions of inverse SL problem by using different input data as eigenvalues and nodes (zeros of eigenfunctions), separately, and calculate the computed errors related to the obtained approximate solutions. To calculate the approximate solutions, we use Chebyshev interpolation technique by applying first kind of Chebyshev polynomials. Eventually, the numerical results are presented by providing some examples.

Restoration of a potential from noisy spectral data

Interest in the inverse Sturm–Liouville problem is motivated by its numerous applications in mathematics and computational physics. To solve a complete inverse problem, one needs two exact spectra, which are usually not known in experimental spectroscopy. Accordingly, a problem of interest is to restore the potential from a finite set of noisy spectral data. A new variational method for solving inverse spectral problems is proposed, which is based on the regularization of ill-posed problems. The method takes into account the measurement error of the spectrum and restores the potential without using simplifying assumptions that it belongs to a certain functional class. The method has been tested on potentials involving smooth segments and jump discontinuities.

A new kind of uniqueness theorems for inverse Sturm-Liouville problems

We prove Marchenko-type uniqueness theorems for inverse Sturm-Liouville problems. Moreover, we prove a generalization of Ambarzumyan’s theorem.

Numerical Reconstruction of Potentials Based on a Sequence of Inverse Sturm–Liouville Problems

Numerical solutions to inverse Sturm–Liouville problems by three spectra, on a finite interval [Formula: see text] are considered. Dirichlet boundary conditions are assumed to hold at the end-points of the interval [Formula: see text], and either a Dirichlet or a Robin-type interior point condition is imposed at an arbitrary interior point [Formula: see text] of [Formula: see text]. We demonstrate that the numerical solution of the inverse three spectra problem corresponding to an interior point [Formula: see text] depends continuously on [Formula: see text]. This statement is particularly useful when measurements of the input data cannot be obtained at the interior point [Formula: see text], but they can be obtained at a nearby interior point. Consequently, we could solve approximately the original inverse problem by departing slightly from [Formula: see text]. We validate this statement through numerical experiments for sequences of interior points [Formula: see text] converging to [Formula: see text].

On a modified Numerov's method for inverse Sturm–Liouville problems

ABSTRACTIn this paper, we discuss the convergence of modified Numerov's method in Q. Gao, X.L. Cheng and Z.D. Huang [Modified Numerov's method for inverse Sturm–Liouville problems, J. Comput. Appl. Math. 253 (2013), pp. 181–199] for computing symmetric potentials from finite Dirichlet eigenvalues. A sufficient condition for convergence of the estimate to the true potential is given and the rate of convergence is investigated. The proof relies on the asymptotics of eigenvalues of the Sturm–Liouville operator and the errors in the finite difference eigenvalues obtained by Numerov's approach. Some numerical experiments are presented to confirm the theoretically predicted convergence properties.

On Inverse Sturm-Liouville Problems With Symmetric Potentials

In this paper an inverse problem − y ″ + p x y = λ y y ′ 0 − h y 0 = y ′ 1 + h y 1 = 0 where p are integrable on [0, 1] and satisfy the symmetry conditions p (x ) = p (1 − x ) almost everywhere in the interval 0 ≤ x ≤ 1. We reconstruct the potential of the inverse Sturm Liouville problem's solution is obtain for finite interval with symmetric potential.

Three spectra inverse Sturm–Liouville problems with overlapping eigenvalues

Three spectra inverse Sturm–Liouville problems with overlapping eigenvalues

Uniqueness results for inverse Sturm-Liouville problems with partial information given on the potential and spectral data

We consider the inverse spectral problem for a Sturm-Liouville problem on the unit interval $[0,1]$ . We obtain some uniqueness results, which imply that the potential q can be completely determined even if only partial information is given on q together with partial information on the spectral data, consisting of the spectrum and normalizing constants. Moveover, we also investigate the problem of missing both eigenvalues and normalizing constants in the situation where the potential q is $C^{2k-1}$ near a suitable point.

Uniqueness of reconstruction of the Sturm–Liouville problem with spectral polynomials in nonseparated boundary conditions

Uniqueness theorems for solutions of inverse Sturm–Liouville problems with spectral polynomials in nonseparated boundary conditions are proved. As spectral data two spectra and finitely many eigenvalues of the direct problem or, in the case of a symmetric potential, one spectrum and finitely many eigenvalues are used. The obtained results generalize the Levinson uniqueness theorem to the case of nonseparated boundary conditions containing polynomials in the spectral parameter.

Inverse spectral problems for differential operators on spatial networks

A short survey is given of results on inverse spectral problems for ordinary differential operators on spatial networks (geometrical graphs). The focus is on the most important non-linear inverse problems of recovering coefficients of differential equations from spectral characteristics when the structure of the graph is known a priori. The first half of the survey presents results related to inverse Sturm–Liouville problems on arbitrary compact graphs. Results on inverse problems for differential operators of arbitrary order on compact graphs are then presented. In the conclusion the main results on inverse problems on non-compact graphs are given. Bibliography: 55 titles.

Functional Equations Related to Sine Type Functions

Functional equations in three classes of entire functions related to sine type functions are considered. These functional equations occur in inverse Sturm–Liouville problems with eigenvalue parameter dependent boundary conditions. In the first class \(\mathcal {A}\), which consists of certain pairs of entire functions (P, Q) with leading terms \(-\lambda \sin \lambda a\) and \(\cos \lambda a\), respectively, it is shown that, for each \(\eta \in \mathbb {R}\), the functional equation \(Q(\lambda )Y(\lambda )-P(\lambda )Z(\lambda )=\eta \) has a unique solution (Y, Z) in \(\mathcal {A}\). As a consequence, the functional equation \(Q(\lambda )S_1(\lambda )-P(\lambda )S_0(\lambda )=1 \) has a unique solution \((S_1,S_0)\) in a class of functions with leading terms \(\cos \lambda a\) and \(\lambda ^{-1}\sin \lambda a\), respectively. A third class of families \(\Phi _\alpha \), \(\alpha >0\), \(\alpha \not =1\), consists of entire functions with leading terms \(-\lambda \sin \lambda a+i\lambda \alpha \cos \lambda a\). It is shown that for each \(\phi \in \Phi _\alpha \) and \(\Delta \in \mathbb {R}\), the functional equation \(\phi (\lambda )X(-\lambda )-\phi (-\lambda )X(\lambda )=2i\alpha \lambda \Delta \) has a unique solution \(X\in \Phi _\alpha \).

On the solvability of inverse Sturm–Liouville problems with self-adjoint boundary conditions

Theorems on the existence and uniqueness of a solution of the inverse Sturm–Liouville problem with self-adjoint nonseparated boundary conditions are proved. As spectral data two spectra and two eigenvalues are used. The theorems generalize the Levitan–Gasymov solvability theorem and Borg’s uniqueness theorem to the case of general boundary conditions.

Uniqueness theorem for Inverse Sturm–Liouville Problem with Nonseparated Boundary Conditions

Consider the string, which vibrates in a medium with the variable elasticity coefficient q(x). Interesting to follow the inverse problem: is it possible to determine the variable elasticity coefficient q(x) by the natural frequencies of string vibrations. In 1946, G. Borg has been shown that a spectrum of frequencies is not sufficient to uniquely identify the medium elasticity coefficient q(x). He offered the use of two frequency spectrum to uniquely identify of the medium elasticity coefficient q(x). The second frequency spectrum is obtained by fastening the string to change at one of its ends to the other fastening. It was shown that these two frequency spectra already sufficient to uniquely identify q(x) and the boundary conditions of both problems. The case where the string fastening at one end depends on the other end fastening, is more difficult to solve. The boundary conditions, appropriate for the occasion, called nonseparated. Two spectra (of two boundary value problems) to restore both q(x), and the nonseparated boundary conditions are not enough. In modern studies the spectra of the two eigenvalues boundary problems and an infinite sequence of signs is generally used for an uniqueness recovery. While this approach is useful in theoretical mathematics, it is inconvenient for the mechanics, because not clear the physical meaning of the corresponding sequence of signs. In this article, instead of the two spectra and the sequence of signs as the spectral data are offered to use 7 of the eigenvalues of the initial boundary value problem, the spectrum, and the so-called norming constants of other boundary value problem. The physical sense of these data is quite clear. The first 7 eigenvalues of an initial boundary problem mean the first 7 natural frequencies of string vibrations. Norming constants represent norms from eigenfunctions. The spectrum and norming constants express a so-called spectral function. The spectral function gives a frequency spectrum with columns of vibrations amplitudes characteristics for string vibrations with other types of fastening.